Modeling time varying and multivariate environmental conditions for extreme load prediction on offshore structures in a reliability perspective

MODELING TIME VARYING AND MULTIVARIATE ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD PREDICTION ON OFFSHORE STRUCTURES IN A RELIABILITY PERSPECTIVE ZHANG YI NATIONAL UNIVERSITY OF SINGAP

MODELING TIME VARYING AND MULTIVARIATE

ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD

PREDICTION ON OFFSHORE STRUCTURES IN A

RELIABILITY PERSPECTIVE

ZHANG YI

NATIONAL UNIVERSITY OF SINGAPORE

2014

MODELING TIME VARYING AND MULTIVARIATE

ENVIRONMENTAL CONDITIONS FOR EXTREME LOAD

PREDICTION ON OFFSHORE STRUCTURES IN A

RELIABILITY PERSPECTIVE

ZHANG YI

( B Eng Nanyang Technological University )

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2014

To my mother

I would like to thank Dr Zhang Mingqiang for his generously sharing with

me his knowledge on the uncertainty modeling and experience of Ph.D studies I could never adequately express all the helps and supports that he has given to me

I like to share my joy of completing the thesis with my friends, especially

Dr Zhang Zhen, Miss Liu Mi, Miss Ge Yao, Mr Dai Jian, Dr Wang Yanbo, Dr Wang Li, Dr Ye Feijian, Mr Lu Yitan and Mr Luo Min I also want to extend my sincere thanks to the other colleagues from the structural lab of National University of Singapore Their helpful discussion and persistent friendship has made my Ph.D study quite enjoyable and fruitful

My deepest gratitude goes to my family for their help, encouragement and support through all these years Most importantly, I owe my loving thanks to my mother for her unlimited love and warm care Without her continuous support and encouragement, I never would have been able to achieve my goals

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Finally, I wish to acknowledge the heartwarming support provided by my dear wife, who has always been the person most understand me and give me the unflagging love

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TABLE OF CONTENTS

TITLE PAGE

DECLARATION PAGE

ACKNOWLEDGEMENT i

TABLE OF CONTENTS iii

SUMMARY vii

LIST OF TABLES x

LIST OF FIGURES xiii

LIST OF SYMBOLS xvii

Chapter 1 Introduction 1

1.1 Background 1

1.1.1 Robust Extreme Models 4

1.1.2 Time Varying Environment 5

1.1.3 Multivariate Environment 6

1.1.4 Efficient Methods for Multivariate Analysis 7

1.2 Objectives and Scope of Thesis 8

1.3 Limitations 9

1.4 Organization of Thesis 10

Chapter 2 Literature Review 14

2.1 Environment Modeling in Analysis of Offshore Structures 14

2.2 Framework of Reliability Analysis 19

2.2.1 Measures of Reliability 21

2.2.2 Simulation Methods 22

2.2.3 Transformation Techniques 26

2.3 Long Term Assessment Criteria 28

2.4 Extreme Value Theory 32

2.4.1 Asymptotic Model 33

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2.4.2 Inference for the Extreme Value Distribution 35

2.5 Concluding Remarks 39

Chapter 3 Establishing Robust Extreme Value Model 41

3.1 Introduction 42

3.2 Peak-Over-Threshold (POT) Method 44

3.2.1 Pareto Family 44

3.2.2 Poisson-GPD Model 45

3.2.3 Declustering 48

3.2.4 Parameter Estimate Method 53

3.3 Uncertainty Assessment of POT Method 57

3.3.1 Effects of Tail Behavior 59

3.3.2 Effects of Noise 64

3.3.3 Effects of Range of Dependency 70

3.4 Effects of Nonstationarity through Random Set Approach 76

3.4.1 Review of Random Set and Dempster-Shafer Structure 77

3.4.2 Selection of Threshold and Time Span 80

3.4.3 Uncertainty Quantification 85

3.5 Concluding Remarks 89

Chapter 4 Modeling the Time Varying Environmental Condition for Offshore Structural Analysis 92

4.1 Introduction 93

4.2 Field Data at Ocean Site 96

4.2.1 Seasonal Characteristics 96

4.2.2 Directional Characteristics 98

4.3 Test for Stationarity of Poisson-GPD model 103

4.3.1 Segmentation Algorithm for Seasonality 104

4.3.2 Segmentation Algorithm for Directionality 113

4.4 Time Varying Modeling 118

4.4.1 2D Fourier Series Characterization 118

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4.4.2 Model Validation 123

4.5 Static Push-Over Analysis 129

4.5.1 Structural Model Description 129

4.5.2 Reliability Analysis with Importance Sampling 132

4.6 Concluding Remarks 139

Chapter 5 Modeling the Multivariate Environmental Condition for the Offshore Structural Analysis 142

5.1 Introduction 142

5.2 Bivariate Models for Sea State Parameters 145

5.2.1 Conditional Joint Distribution Model 145

5.2.2 Nataf Model 148

5.3 Copula Theory 150

5.3.1 Definition and Basic Properties 150

5.3.2 Examples of Copula 153

5.3.3 Dependence Concepts 158

5.4 Comparative Study in Multivariate Modeling 160

5.4.1 Data Pre-treatment 160

5.4.2 Application of Bivariate Models 167

5.4.3 Results and Discussions 175

5.5 Time Domain Structural Analysis 180

5.5.1 Proposed Discretized Copula Approach 182

5.5.2 Structural Analysis of a Fixed Offshore Platform 187

5.5.3 Results and Discussions 196

5.6 Concluding Remarks 200

Chapter 6 Conclusions 202

6.1 Summary of Thesis 202

6.2 Recommendation of Future Works 204

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REFERENCES 209

Appendix A Detailed Information of Four Discretization Steps 223

Appendix B Uncertainty Assessment in POT Method 239

Appendix C Selection of Threshold and Time Span in POT Method 254

Appendix D Testing Values of Threshold and Time Span in POT Approach for Each Identified Time Sectors 259

Appendix E USFOS Program Input for the Example Structure 264

Appendix F Example of Constructing a Random Set Model 285

Appendix G Information of Selected Wave Data and Basic Linear Wave Theory 288

This thesis focuses on the characterization of the time varying characteristics of variables associated with the environmental loads, the dependencies between these variables, and investigating the impact of the uncertainties and dependencies on the long term assessment of a typical marine structure The wave parameters which directly influence the loadings on offshore structures are studied in this work

Considering the long time life-span and the limited data normally available, extreme value (EV) statistical models have been commonly adopted to describe environmental loads Such class of models has assumptions that may not

be fulfilled for some variables, such as independence between data points and the stationarity of the data As such, compound extreme value statistical models have also introduced, for example, the peak-over-threshold (POT) model Using the

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same set of simulated data, both the EV and POT models are compared in this study, where the latter is found to be a more inclusive and better approach It is also found that the accuracy associated with each model is sensitive to the parameter estimate method used, especially if available data is limited To characterize the uncertainties associated with the parameters, namely, the

threshold u and time span ∆t, a random set model is proposed in the context of the

POT approach Such elaborate measure in the form of imprecise probabilities could reflect the intensities of uncertainty in the selection of parameters more realistically

By analyzing various segments of the collected data, stationarity of the parameters can be established If the non-stationary is slowly time varying, then a simple means to account for this is through appropriate division of the data into stationary segments A modified segmentation algorithm with specified fixed time interval is proposed to extract homogeneous data sets from non-stationary time series data To select the time interval, the data fitting is recursively performed until the sample data can satisfactorily fit the Poisson-Generalized Pareto Distribution (GPD) model Two-dimensional Fast Fourier transform is employed to characterize the variations of extreme values with time A collected group of data is selected to demonstrate that such a discretized model can provide

a more reasonable and accurate characterization of each parameter of interest This approach of incorporating the time varying effect is examined through the reliability analysis of an existing offshore platform The results show that incorporating the co-variate effects in the statistical model can better reflect the

as the most critical sea states are identified The newly developed method can be extended to a multivariate problem based on the copula model

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LIST OF TABLES

Table 2.1 Distribution models for selected ocean parameters 16

Table 2.2 Target reliability levels in design codes 30

  Table 3.1 Bias of estimated 99th percentile for different tails n=20, 100 63

Table 3.2 Bias of estimated 99th percentile for different noise condition n=20, 100 70

  Table 4.1 Statistics of H S over four defined seasons 97

Table 4.2 Statistics of wave directions over four defined seasons 99

Table 4.3 Summary of tested algorithms in different segmentations of H S with respect to time 111

Table 4.4 Summary of tested statistics for different segmentations of extreme H S with respect to directionality 117

Table 4.5 Estimated shape parameter ξ in each discretized sector 117

Table 4.6 Estimated scale parameter σ in each discretized sector 117

Table 4.7 Random variables in reliability analysis 132

Table 4.8 Determined coefficients of Eq (4.19) for different directions 134

Table 4.9 Failure probability of jacket structure for different time and directional sector of wave model 136

Table 4.10 Determined coefficients of Eq (4.19) for rotated structure 138

Table 4.11 Failure probability of rotated jacket structure with respect to wave model 139

  Table 5.1 Conditional joint distribution models for selected stochastic ocean parameters 147

Table 5.2 Examples of Archimedean Copulas 155

Table 5.3 Regression results for points representing H S within interval values of 0.5 168

Table 5.4 Regression results for points representing H S within interval values of 0.1 170

Table 5.5 Regression results for k vs H S and λ vs H S 172

Table 5.6 Results of marginal distribution model parameter estimates 174

Table 5.7 Comparison of parameter estimates and goodness-of-fit to the data for the 3 approaches 176

Table 5.8 Sea states and values of simulation parameters for the same accuracy 196

Table B.1 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =-0.5,

σ=2) 240

Table B.2 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =-0.25,

σ=2) 241

Table B.3 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =0,

σ=2) 242

Table B.4 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =0.25,

σ=2) 243

Table B.5 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =0.5,

σ=2) 244

Table B.6 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1+N(0,0.1),

ξ =-0.5, σ=2) 245

Table B.7 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1+N(0,0.3),

ξ =-0.5, σ=2) 246

Table B.8 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1+N(0,0.5),

ξ =-0.5, σ=2) 247

Table B.9 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =-0.5,

σ=2+N(0,0.2)) 248

Table B.10 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =-0.5,

σ=2+N(0,0.6)) 249

Table B.11 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ =-0.5,

σ=2+N(0,1.0)) 250

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Table B.12 Bias of estimated results (shape parameter ξ, scale parameter σ and

99th percentile) based on simulations from GPD model (u=1, ξ 0.5+N(0,0.05), σ=2) 251 Table B.13 Bias of estimated results (shape parameter ξ, scale parameter σ and

=-99th percentile) based on simulations from GPD model (u=1, ξ 0.5+N(0,0.15), σ=2) 252 Table B.14 Bias of estimated results (shape parameter ξ, scale parameter σ and

=-99th percentile) based on simulations from GPD model (u=1, ξ 0.5+N(0,0.25), σ=2) 253

=- 

Table C.1 P-value of K-S test for exceedances following GPD with different

values of (u, ∆t) 254

Table C.2 P-value of K-S test for occurrences of exceedances following Poisson

process with different values of (u, ∆t) 255

Table C.3 Estimated lower bound of 0.95 confidence interval for 100 year design

value with different values of (u, ∆t) 256

Table C.4 Estimated upper bound of 0.95 confidence interval for 100 year design

value with different values of (u, ∆t) 257

Table C.5 Number of exceedances above the threshold after applying the POT

method with different values of (u, ∆t) 258

Figure 2.1 Failure domain of performance function G(.) in the variable space 21

Figure 2.2 Schematic showing of statistical response characterization in reliability

analysis 25 Figure 2.3 Reliability analysis by using response surface method 28

Figure 2.4 Illustration of different tail behavior in Type I (ξ =0, μ=0, σ=1), Type

II (ξ =0.5, μ=0, σ=1) and Type III (ξ =-0.5, μ=0, σ=1) 35

Figure 3.1 (a) Scatter plot of time series measurement (b) Clusters identification

(c) Identified extreme values after declustering 50 Figure 3.2 Bias of estimated scale parameter for different tails 62 Figure 3.3 Bias of estimated shape parameter for different tails 63 Figure 3.4 Bias of estimated scale parameter ((a), (b) and (c)) and shape

parameter ((d), (e) and (f)) with the noise effect in location parameter

in GPD model 67 Figure 3.5 Bias of estimated scale parameter ((a), (b) and (c)) and shape

parameter ((d), (e) and (f)) with the noise effect in scale parameter in GPD model 68 Figure 3.6 Bias of estimated scale parameter ((a), (b) and (c)) and shape

parameter ((d), (e) and (f)) with the noise effect in shape parameter in GPD model 69 Figure 3.7 Plots of autocorrelation functions for two tested time series (a) Case 1:

φ=0.95 (b) Case 2: φ=0 71

Figure 3.8 Biases of estimated 99th percentile in AMM and r largest order statistic

method for two time series (a) Case 1: φ=0.95 (b) Case 2: φ=0 73

Figure 3.9 Biases of estimated 99th percentile in POT method for two time series

(a) Case 1: φ=0.95 (b) Case 2: φ=0 74

Figure 3.10 Imprecise probability 80 Figure 3.11 Mean residual plot with 95% confidence intervals 82 Figure 3.12 L-moment plot for exceedances over selected threshold with

theoretical GPD curve 83 Figure 3.13 Autocorrelation plot for the wave time series records 84

Figure 3.14 Appropriate region for u and ∆t 85

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Figure 3.15 Constructed imprecise probability model for 100-year return value

with mean value of all estimates 87

Figure 3.16 Constructed imprecise probability model for 100-year return value based on different sample set: (a) Jan (b) Jan-Mar (c) Jan-Jun Dotted line represents mean of all estimates 89

  Figure 4.1 Time varying effects on safety of structures 94

Figure 4.2 Box plot of H S for four different defined seasons 97

Figure 4.3 Histograms of H S in four different defined seasons 98

Figure 4.4 Angular histograms of H S over four different defined seasons (red line represents the resultant vector length) 100

Figure 4.5 2D kernel density plot of H S with directionality and seasonality 101

Figure 4.6 Isoline plot of H S with directionality and seasonality 102

Figure 4.7 Results of t-statistic for discretized time series 108

Figure 4.8 2D kernel density plot of extreme H S with directionality and time 112

Figure 4.9 Comparison of C i values between different segmentations for time sector [0.10, 0.35) 116

Figure 4.10 Discrete spectrum for shape and scale parameters after DFT 121

Figure 4.11 2D Fourier characterizations of shape and scale parameter changes along the time and direction axes 121

Figure 4.12 CDF of original data and simulated data for θ (-10o~62o] 124

Figure 4.13 CDF of original data and simulated data for θ (62o~134o] 125

Figure 4.14 CDF of original data and simulated data for θ (134o~206o] 125

Figure 4.15 CDF of original data and simulated data for θ (206o~278o] 126

Figure 4.16 CDF of original data and simulated data for θ (278o~350o] 126

Figure 4.17 Comparison of simulated and original mean of H S with time 128

Figure 4.18 Comparison of simulated and original observed wave directions 128

Figure 4.19 Jacket structure framing 129

Figure 4.20 Plasticity utilization plot 131

Figure 4.21 Comparison between calculated base shear and curve fit 135

  Figure 5.1 Sketch of several conditional probability density curves of T P at various values of H S 146

Figure 5.2 Comparison of different bivariate copulas for marginal distributions following standard normal distributions for correlation coefficient equals to 0.8 156

Figure 5.3 Box plot of H S and V W over different months 161

Figure 5.4 Comparison of dependencies between H S and T P for two different months 164

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Figure 5.5 Comparison of dependence between θ S and θ W over two different

months 165 Figure 5.6 Rose plot of wave direction in Nov-Feb 166 Figure 5.7 Histogram of difference between wind direction and wave direction

(θ S - θ W) 166

Figure 5.8 Nonlinear fit of (a) μ[ln(T P )] vs H S and (b) σ2[ln(T P )] vs H S based on

Eq (5.1) and Eq (5.3) for points representing H S within interval

values of 0.5 168

Figure 5.9 Nonlinear fit of (a) μ[ln(T P )] vs H S and (b) σ2[ln(T P )] vs H S based on

Eq (5.1) and Eq (5.3) for points representing H S within interval values of 0.1 170

Figure 5.10 Nonlinear fit of (a) k vs H S and (b) λ vs H S based on Eqs (5.23) &

(5.24) 171

Figure 5.11 Marginal parametric model fit for (a) H S , (b) T P , and (c) V W 173 

Figure 5.12 Tail fittings of marginal parametric model for (a) H S , (b) T P, and (c)

V W 174 

Figure 5.13 Comparison of contour plot between original data and (a) copula

approach, (b) Nataf model, (c) conditional joint model for H S and T P 178 Figure 5.14 Comparison of contour plot between the original data and (a) copula

approach (b) Nataf model (c) conditional joint model for H S and V W 179 Figure 5.15 Schematic showing of proposed discretization procedure 187 Figure 5.16 Illustration of JONSWAP spectrum for different sea states 189 Figure 5.17 Illustration of JONSWAP spectrum for different peak shape

parameters 190

Figure 5.18 Cut off frequency ω cut for different peak shape parameters 192

Figure 5.19 Determination of number of frequencies N for a given peak period.

193

Figure 5.20 Determination of time interval ∆t for a given peak period 193

Figure 5.21 Typical simulated wave elevations 195

Figure 5.22 Schematic showing of discretization steps (u=F1(H S ), v=F2(T P)) and

evaluation points 197   

Figure 6.1 Schematic showing of discretization procedures for three-dimensional

copula model 206

Figure A.1 Numbering of subcopulas in first discretization 233 Figure A.2 Numbering of subcopulas in second discretization 234

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Figure A.3 Numbering of subcopulas in third discretization 234 Figure A.4 Numbering of subcopulas in fourth discretization 235 Figure A.5 Comparison of structural base shear for different sea states 238   

Figure C.1 Appropriate region for u and ∆t for significance level equals to 0.01.

255

Figure C.2 Influence of u and ∆t to (a) scale parameter, (b) shape parameter and (c)

100 year return value 257

Figure G.1 WIS data domain 288 

Figure G.2 Geological location of the selected buoy 289 

Figure G.3 Regular wave propagation properties 290

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LIST OF SYMBOLS

C Archimedean(.) Archimedean copula

CDF cumulative distribution function

C Gaussian(.) Gaussian copula

d differentiation

DFT discrete Fourier transformation

D i cumulative sum statistic for Poisson process

Exp(.) exponential distribution function

upper bound of cumulative probability function lower bound of cumulative probability function

CDF of annual maximum of 10-minute mean wind speed

CDF of maximum wave height in a stationary sea state

conditional CDF of 1-hour mean wind speed for a

CDF of short term distribution of wave height

F ξ , F σ values of discrete spectrum for shape and scale

parameters

cumulative distribution functions for sector S i

| conditional PDF of T p for a given value of H S

GEV generalized extreme value distribution

GPD generalized Pareto distribution

Gr(.) r largest order statistics distribution function

H 100max maximum wave height in 100 years

I x(a, b) incomplete beta function

rth largest value in a sequence of random variables

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N number of simulated wave frequency

PDF probability density function

R ultimate ultimate strength of jacket structure

t i seasonality value for observed wave height

U vector of standard normal variables

horizontal water particle acceleration

θ i directionality value for observed wave height

dependency

ρ θ angular linear correlation coefficient

σ x standard deviation of variable x

τ k Kendall’s coefficient for measuring the dependency (.) probability density function for standard normal

∆t time_step time step in time domain analysis

∆T time interval in the time segmentation algorithm

Chapter 1 Introduction

1.1 Background

Offshore structures facilitate the exploitation of the vast ocean resource which contributes significantly to technological and economic development Since the first oil platform started operating at the ocean of Couissana in 1947, thousands of marine structures have been built over the last six decades Up to now, there are more than 4000 platforms operating at the Gulf of Mexico alone The evolution

of marine technology is rapid and numerous kinds of structures are now available

in the open sea, encompassing fixed structures (for example, jackets and jack-ups)

in shallow waters and floating structures (for example, tension leg platforms and semi-submersibles) in deep waters, see Fig 1.1

Figure 1.1 Different types of offshore structures (Roy 2013)

Compared with normal structures on land, offshore structures are bulky, expensive and in some cases have complex geometry The marine environment for offshore structures can be severe, adverse, varied and uncertain It covers a broad area of climatic factors which generally includes wind, waves, current, ice, tide and other catastrophe events such as earthquake, storm and tsunami Under such environment, many unfavorable phenomena like marine corrosion, marine growth, foundation scouring, material deterioration and fatigue damage will cause

a weakening of the overall strength of the structure and thus lead to an unexpected accident Historical records depicted our inadequate understanding of the ocean environment leading to accidents with drastic consequences and huge economic loss Examples include the first UK-built semi-submersible rig Ocean Prince and the first rig to find oil in UK waters, which broke up off England's east coast during a storm in 1967 (Oo 1974) Bohai No 2 jack-up structure located in the Gulf of Bohai between China and Korea encountered a storm and sank on 25 November 1979, resulting in the deaths of 72 out of the 74 personnel on board (Santos and Feijo 2010) In 1980, the semi-submersible accommodation rig Alexander L Kielland capsized during a storm after the brace supporting a leg failed in the Norwegian Continental Shelf (BMT 2006) An example of recent failures is the Usumacinta jack-up in the Gulf of Mexico which was struck by a strong storm on 21 October 2007, resulting in a fatal blowout causing 21 reported deaths with one worker missing during the evacuation (OGP 2010), see Fig 1.2 These painful lessons provide strong motivation to improve our knowledge of the

climate-dependent ocean and its effects so as to enhance the safe operation of offshore structures

Figure 1.2 Controlled burning in Usumacinta jack-up after its first fire (OGP 2010)

To enforce the safety requirements imposed by design codes (such as Ultimate Limit State (ULS) and Fatigue Limit State (FLS) criteria to achieve a target level of safety not exceeding more than once in 100 years (DNV 2007)) are followed, an assessment of the structure under consideration must be carried out The objective is to ensure safe performance and limit fatalities as well as damages caused by the environment and operational loads during its service life

Various methods for safety assessment of existing offshore structures have been developed in relation to severe environmental loading such as hurricanes (Moan 2005) The results from various methods are very much dependent on the validity of the assumptions imposed (Fitzwater and Winterstein 2001; Moriarty et

al 2006; Saranyasoontorn and Manuel 2006; Naess et al 2007) The design code recommends several statistical models to represent physical variables such as wind and waves Due to the complexity of the environment, these models may not be able to capture the complete characteristics of the physical variables For example, the wave load which constitutes roughly 80% of the overall external load, is the major environmental loads The load is affected by factors such as its period, time and direction of loading, that a single statistical model could hardly

be accurate in describing the randomness of the waves Obtaining an appropriate wave load model remains a challenge The model should be sufficiently robust to

be able to represent the most critical situation and produce a good estimate of the reliability of the structure under investigation

1.1.1 Robust Extreme Models

The traditional modeling based on probabilistic theory has been well established and used in the characterization of environmental factors in the offshore industry (Muir & El-Shaarawi 1986) The parameters governing the models are estimated statistically using data collected from monitoring stations Often only the extreme values govern the design loads and the design life span becomes a primary determining variable

A commonly adopted approach is to use extreme value statistical models

to arrive at a design value corresponding to the structure’s lifespan The direct annual maximum method, which models the largest value from each year, is most

widely adopted (Gumbel 1958; Leadbetter et al 1983) However, significant inaccuracies or misrepresentation may occur if only limited amount of data are available for constructing a statistical model Usually, the length of field data is limited to several decades which are considered short in relation to the design life

of the structure of interest Consequently, other approaches have been proposed, such as the Peak-Over-Threshold (POT) method, which characterizes the exceedance over a high threshold, to utilize more data in forming the extreme value model

Even with such methods as POT, several important associated questions have not been addressed These include issues on the choice of the parameter estimation method, the applicability of the method for data sample size that is small, the effect of serial dependencies, the effect of noise in the observed time series data, and the quantification of uncertainties associated with the selected threshold and limited time span of the data Not all these issues have been adequately addressed in the literature (Næss 1998; Mackay et al 2010; Jocković 2013)

1.1.2 Time Varying Environment

Based on different channels of observations, it is increasingly evident that the effects of climate change, especially with regards to the ocean environment, cannot be ignored The contention is whether the assumption that the averages and extremes of sea states are stationary is valid (Vanem 2011) If non-

stationarity is assumed, a simple statistical model for the ocean parameter may not

be appropriate in assessing the safety of a structure under such environment Using the classical extreme statistics approach for climate-induced time varying phenomena may lead to bias estimations for the parameters of the extremal models and in turn, may have drastic consequences It is important to derive a statistical model that could reflect the changes in the value with time to ensure that the reliability of the structure is estimated with minimal error Recent researches showed that a time-dependent version of extreme value model could help to capture this non-stationary characteristic in the ocean data (Méndez et al 2006; Marcos et al 2011) This has sparked interest in incorporating such model in the safety assessment of marine structure over its design life (Menendez et al 2009), in particular accounting for all the uncertainties associated with such an approach (Vanem and Bitner-Gregersen 2012)

1.1.3 Multivariate Environment

It is natural that the environment parameters, such as wave period, significant wave height and wind speed, are correlated To assume that they are independent may lead to unconservative results and hence is a potential cause for under-design leading to possible catastrophic consequences Multivariate statistical model would be an appropriate tool to handle such complications There exist practical guidelines on the choice of joint distribution models to characterize a multivariate environment (DNV 2007) The issue is whether these models can be applied to

all the ocean sites The suitability of various computational methods to handle dependencies has yet to be completely answered It is therefore of interest to perform a comparative study highlighting the characteristics in each approach in modeling multivariate data

1.1.4 Efficient Methods for Multivariate Analysis

To assess the long term performance of a structure within a multivariate environment associated with many environment conditions requires many numerical simulations and is computationally very expensive The current practice is to divide the scatter diagram describing the environmental parameters into blocks and to calculate the response of the structure associated with for each block The results for all blocks are then combined to obtain the overall response distribution Such an approach involves numerous combinations even with a small number of environment conditions, not to say that the computation of the response for each combination can be computationally cumbersome A complete evaluation may demand hundreds of hours of CPU time on a personal computer (Agarwal and Manuel 2009) Besides, it may not be easy to deduce from such computational results the critical environment conditions that contribute significantly to the overall long term performance of the structure (for example, floating structures could be quite sensitive to wave loading pertaining to certain critical wave periods and/or directions) It is thus of interest to develop efficient

methods, including one that allows for quick sensitivity studies to deduce the significant parameters and their range of critical values

1.2 Objectives and Scope of Thesis

As a further contribution towards a more realistic assessment of the reliability of offshore structures, the main objective of the proposed research is to develop an accurate statistical model and reliability computation procedure which will consider the time varying uncertain non-stationary characteristics of the wave loads, as well as the dependencies amongst the key parameters To achieve this objective, the main foci of the research are:

1 To study the uncertainties related to an existing extreme value model The aim

is to have an improved understanding of the importance of the methods selected for the construction of an extreme value model The performances of the methods are compared for different types of data groups In addition, the concept of random set to tackle the difficulties associated with parameter selections in the Peak-over-Threshold extreme method is presented

2 To manage the non-stationarity inherent in long term data by developing a discrete statistical model to represent the time varying effects in the ocean parameters and show the importance of this in a reliability analysis A segmentation approach is employed and 2-D Fourier transforms is used in conjunction with a probabilistic model to reflect the smooth transition in the parameters amongst the segments The estimated reliability associated with

each segment can be studied to ascertain whether effects such as seasonality and directionality are significant

3 To provide a better model for the multivariate environment beyond the linear correlation coefficient representation Specifically, the discretized copula approach is introduced and evaluated for it suitable application towards the structural reliability of offshore structures by comparing the results with other existing multivariate models The computational efficiency of the proposed technique is discussed

1.3 Limitations

While this work endeavor to develop statistical procedures for accurate modeling

of environmental effects for offshore design, many simplifications were made in the reliability analysis The load model uncertainty, including the wave kinematics and hydrodynamic load, for a jacket structure are not considered The current analysis only focus on the ultimate limit state of jacket structure and the complex behavior of individual elements in the structure will not be analyzed in detail The soil foundation is simplified to act like a spring in the structural analysis With regards to the environment loads, only wave loads are considered using an idealized random linear wave model Besides, the effects of breaking wave, the coupling between the wave and structure, wave and current are not addressed in this study Rather, the key focus is on structural reliability analysis

by considering environmental statistical model The effects of corrosion, fatigue

damage and deterioration which are not likely to occur in a short time are neglected in the structural model The developed procedures in handling the field data serves as a pre-cursor step to a full numerical analysis of an offshore engineering problem The investigated statistical model in this thesis is only based

on the collected data which is independent of the structural types Therefore, other types of offshore structures can also be applied in this case Lastly, the results and findings of this study is based on limited data from a given geological area The models adopted in this study may not be the most appropriate ones in every case, especially if the field data is taking from one place that is different from the chosen site

The conclusions drawn from the thesis should be seen in the light of these limitations The influence of these limitations to the reliability results may need further investigations in the future

1.4 Organization of Thesis

This thesis is organized into six chapters as follows

Chapter 1 outlines the motivation for the research, leading to the proposed research objectives and methodology This chapter addresses several issues in performing the reliability analysis of offshore structures under it operational environment Three key topics associated with the long term safety assessment of offshore structures are highlighted and forms the research topics presented in the subsequent chapters

Chapter 2 presents a concise literature review of the current state of knowledge for the reliability analysis of offshore structures The basic mathematical model of characterizing extreme values is presented Several probabilistic analysis techniques and their applicability in offshore engineering are reviewed, with emphasize on impact of the environment

Chapter 3 discusses the performance of different extreme value approaches in establishing a robust statistical model The importance of parameter estimation method associated with the effects of distribution tail behavior, noise in the data and nature of dependency are investigated To address the difficulties in selecting the threshold and time span in POT approach, a random set based imprecise probability model is proposed to quantify this subjective uncertainty

Chapter 4 presents an improved method for the long term safety assessment of an existing structure where the time-varying characteristics of the environment are incorporated For this purpose, a segmentation algorithm is proposed to improve the quality of the extreme value model for the ocean parameters Comparison of the reliability analysis results from the proposed approach against those from the traditional approach is illustrated using a real structure

Chapter 5 discusses the probabilistic models for multivariate environment variables and the associated reliability estimation method proposed in this thesis

The numerical procedure in evaluating the safety of the structure with environment factors having complex dependency is illustrated

This thesis concludes with Chapter 6 which summarizes the main findings Recommendations of possible future works are provided

Figure 1.3 Organization of the thesis

Chapter 1 Introduction, objective and scope of study

Chapter 2 Literature review, fundamental knowledge

of offshore environmental statistical modeling

Chapter 3 Establishing robust extreme value model

Uncertainty assessment

of POT method

Effects of nonstationarity through random set approach

Chapter 4 Modeling the time

varying environmental condition

for offshore structural analysis

Chapter 5 Modeling the multivariate environmental condition for offshore

structural analysis

Time varying structural

reliability analysis

Modeling multivariate environmental data in copula

Copula based multivariate structural dynamic analysis

Chapter 6 Conclusions and recommendations

Modeling time varying

environmental data by

segmentation

Chapter 2 Literature Review

This chapter presents a review of various techniques related to the reliability assessment of an offshore structure and also the available mathematical models for the ocean environment variables For processing the available information through engineering computations, various aspects through the reliability analysis are discussed such as the required level of accuracy, the ease of handling as well

as the computational efficiency Particularly, the difficulties in assessing an offshore engineering reliability problem are emphasized in this chapter The analysis of the influence of environment variability to the overall structural design

is achieved by structural analysis Moreover, if the long term performance of an offshore structure is a primary concern, the analysis could be realized by the probabilistic approach with the basic concept of extreme value modeling The most key features relevant to the extreme value modeling are discussed in this chapter

2.1 Environment Modeling in Analysis of Offshore Structures

The presence of uncertainties in engineering systems and models has been widely acknowledged (Ang and Tang, 1975; Hokstad et al 1998; Straub and Faber 2005; Gao and Moan 2009; Lee and Song 2011; Agarwal and Manuel 2011) The uncertainties associated with offshore structural analysis lies in the structural

characteristics and associated environmental loadings The former arises from the uncertainty in the response transfer function, the variability of the structure’s strength, geometry as a result of production and manufacturing, and deteriorations

of the materials The latter results from the variability in the environmental factors, such as wind velocity, wave characteristics (magnitude, direction, height and period), and the uncertainty in characterizing the loads

An important task in the design and reliability assessment of an offshore structure is the modeling of the environmental parameters A proper choice of the distribution type of the environmental dependent load parameter is very critical as the results of a reliability analysis may be very sensitive to the tail of the probability distribution However, since there is no theoretical basis to support the choice of any particular model, there are many different methods and models applied to the ocean parameters by various authors The most widely-used distribution for 10-min average wind speeds is the 2-parameter Weibull distribution (Manwell et al 2002; Ramirez and Carta 2005; Morgan et al 2011) Celik (2004) showed that the simple 1-parameter Rayleigh distribution sometimes offers a better fit to the sampled data The Gumbel distribution is usually selected

to model the current velocity (Pugh 1982; Robinson and Tawn 1997; Sauvaget et

al 2000) However, Mazumder and Mazumber (2006) argued that the model should take care of the directional effects Jaspers (1956) was the first to propose

a lognormal distribution model to represent the wave height This was later discussed and compared with many other models, such as the Weibull distribution (Battjes 1972), the mixed Weibull-Lognormal distribution (Haver 1985), the